Theorem elprg 2423

Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized.

Assertion

Ref	Expression
elprg	|- (A e. D -> (A e. {B, C} <-> (A = B \/ A = C)))

Proof of Theorem elprg

Step	Hyp	Ref	Expression
1		eqeq1 1481	. . 3 |- (x = A -> (x = B <-> A = B))
2		eqeq1 1481	. . 3 |- (x = A -> (x = C <-> A = C))
3	1, 2	orbi12d 627	. 2 |- (x = A -> ((x = B \/ x = C) <-> (A = B \/ A = C)))
4		dfpr2 2422	. 2 |- {B, C} = {x | (x = B \/ x = C)}
5	3, 4	elab2g 1900	1 |- (A e. D -> (A e. {B, C} <-> (A = B \/ A = C)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958  {cpr 2410

This theorem is referenced by:  elpr 2424  elpr2 2425  ifpr 2427  elsncg 2430  snsspr 2470  unisn2 2875

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413

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